含有直线裂纹的直角域中椭圆形孔洞对SH波的散射
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摘要
利用Green函数法、复变函数法及保角映射技术研究SH波作用下直角域内直线裂纹对椭圆形孔洞动应力集中系数的影响。采用保角映射法和镜像叠加原理构造一个能自动满足直角平面2个自由表面应力自由边界条件的散射位移场,并取含有椭圆形孔洞的直角域中任意一点承受时间谐和的出平面线源载荷作用下的位移基本解作为适合的Green函数。利用裂纹"切割"技术构造裂纹,求解出裂纹与椭圆形孔洞共存时的位移场和应力场。具体算例中给出椭圆形孔洞动应力集中系数分布情况并对结果进行分析和讨论。结果表明裂纹的长度、方位角、距孔洞的距离等参数均对动应力集中系数的分布有影响。
The scattering problem of SH-wave by a shallowly buried ellipse cavity in right-angle space with a beeline crack was analyzed by using Green's function and complex function method with conformal mapping method. The conformal mapping method and image method were employed here to construct the scattering wave field function,which satisfies the condition of stress free on the straight boundaries. An essential solution to displacement field of an elastic right-angle plane containing an ellipse cavity that any point bore an anti-plane harmonic line source load was taken as Green's function. Beeline crack was constructed with crack-division technique,and expression of displacement and stress field were given while both crack and ellipse cavity exist. The dynamic stress concentration factor(DSCF) on the edge of ellipse cavity was given as an example and the results are analyzed and discussed. Results show that the crack length,angle,distance between the ellipse cavity and beeline crack have an effect on the distribution of DSCF of the ellipse cavity.
The scattering problem of SH-wave by a shallowly buried ellipse cavity in right-angle space with a beeline crack was analyzed by using Green's function and complex function method with conformal mapping method. The conformal mapping method and image method were employed here to construct the scattering wave field function,which satisfies the condition of stress free on the straight boundaries. An essential solution to displacement field of an elastic right-angle plane containing an ellipse cavity that any point bore an anti-plane harmonic line source load was taken as Green's function. Beeline crack was constructed with crack-division technique,and expression of displacement and stress field were given while both crack and ellipse cavity exist. The dynamic stress concentration factor(DSCF) on the edge of ellipse cavity was given as an example and the results are analyzed and discussed. Results show that the crack length,angle,distance between the ellipse cavity and beeline crack have an effect on the distribution of DSCF of the ellipse cavity.
引文
[1]LEE V W,TRIFUNAC M D,ASCE A M.Response of tunnels to incident SH-waves[J].Journal of the Engineering Mechanics Division,1979,8:643–659.
[2]陈志刚,刘殿魁.SH波冲击下浅埋任意圆形孔洞的动力分析[J].地震工程与工程振动,2004,24(4):32–36.(CHEN Zhigang,LIU Diankui.Dynamic response on a shallowly buried cavity of arbitrary shape impacted by vertical SH-wave[J].Earthquake Engineering Vibration,2004,24(4):32–36.(in Chinese))
[3]LEE V W,MANOOGIAN M E.Surface motion above an arbitrary shape underground cavity for incident SH wave[J].European Earthquake Engineering,1995,8(1):3–11.
[4]LEE V W,AMORNWONGPAIBUN A.Scattering of anti-plane(SH)waves by a semi-elliptical hill:I-Shallow hill[J].Soil Dynamic and Earthquake Engineering,2012,52:116–125.
[5]AMORNWONGPAIBUN A,LEE V W.Scattering of anti-plane(SH)waves by a semi-elliptical hill:II-Deep hill[J].Soil Dynamic and Earthquake Engineering,2012,52:126–137.
[6]LUO H,LEE V W,LIANG J W.Anti-plane(SH)waves diffraction by an underground semi-circular cavity:analytical solution[J].Earthquake Engineering and Engineering Vibration,2010,9(3):385–396.
[7]LEE V W,LUO H,LIANG J W.Diffraction of anti-plane SH waves by a semi-circular cylindrical hill with an inside concentric semi-circular tunnel[J].Earthquake Engineering and Engineering Vibration,2004,3(2):249–262.
[8]QI H,YANG J.Dynamic analysis for circular inclusion of arbitrary positions near interfacial crack impacted by SH-wave in half-space[J].European Journal of Mechanics/A Solids,2012,36:18–24.
[9]HASSAN A,SONG T S.Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials[J].Applied Mathematics and Mechanics,2014,35(10):1 261–1 270.
[10]杨在林,许华南,黑宝平.双相介质中椭圆孔洞与裂纹对SH波的散射[J].工程力学,2014,31(10):33–39.(YANG Zailin,XU Huanan,HEI Baoping.Interaction of an elliptical cavity and a crack near bimaterials interface under incident SH-waves[J].Engineering Mechanics,2014,31(10):33–39.(in Chinese))
[11]陈志刚.SH波在界面孔上散射的远场解[J].振动与冲击,2014,33(12):138–144.(CHEN Zhigang.Far field solution to SH-wave scattered by an interface cavity[J].Journal of Vibration and Shock,2014,33(12):138–144.(in Chinese))
[12]史文谱,陈瑞平,张春萍.直角平面内弹性圆夹杂对入射平面SH波的散射[J].应用力学学报,2007,24(1):154–160.(SHI Wenpu,CHEN Ruiping,ZHANG Chunping.Scattering of circular inclusion in right-angle plane to incident plane SH-wave[J].Chinese Journal of Applied Mechanics,2007,24(1):154–160.(in Chinese))
[13]齐辉,杨杰,李宏亮,等.含任意直线型裂纹的直角域中圆柱夹杂对SH波的散射[J].振动与冲击,2011,30(5):208–212.(QI Hui,YANG Jie,LI Hongliang,et al.Scattering of SH-wave by a cylindrical inclusion in right-angle plane with arbitrary beeline crack[J].Journal of Vibration and Shock,2011,30(5):208–212.(in Chinese))
[14]QI H,SHI Y,LIU D K.Interaction of a circular cavity and a beeline crack in right-angle plane impacted by SH-wave[J].Journal of Harbin Institute of Technology,2009,16(4):548–553.
[15]杨在林,闫培雷,刘殿魁.SH波对浅埋弹性圆柱及裂纹的散射与地震动[J].力学学报,2009,41(2):229–235.(YANG Zailin,YAN Peilei,LIU Diankui.Scattering of SH-waves and ground motion by an elastic cylindrical inclusion and a crack in half space[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(2):229–235.(in Chinese))
[16]宋天舒,李冬.压电体中孔边Ⅲ型界面裂纹的动应力强度因子[J].力学学报,2010,42(6):1 219–1 224.(SONG Tianshu,LI Dong.Dynamic stress intensity factor for interfacial cracks of mode III on a circular cavity in piezoelectric bimaterials[J].Chinese Journal of Theoretical and Applied Mechanics,2010,42(6):1 219–1 224.(in Chinese))
[17]杨在林,许华南,黑宝平.SH波上方垂直入射时界面附近椭圆夹杂与裂纹的动态响应[J].岩土力学,2013,34(8):2 378–2 384.(YANG Zailin,XU Huanan,HEI Baoping.Dynamic response of elliptical inclusion and crack neat interface under vertically incident SH-wave from above[J].Rock and Soil Mechanics,2013,34(8):2 378–2 384.(in Chinese))
[18]WANG J H,LU J F,ZHOU X L.Complex variable function method for scattering of plane waves by an arbitrary hole in a porous medium[J].European Journal of Mechanics/A Solids,2009,28:582–590
[19]刘中宪,刘蕾,梁建文.楔形空间中任意形状孔洞对平面SH波的散射:IBEM求解[J].固体力学学报,2014,43(3):292–301.(LIU Zhongxian,LIU Lei,LIANG Jianwen.Solving the scattering of plane SH-waves by an arbitrary shaped cavity embedded in a wedge-shaped space by IBEM[J].Chinese Journal of Solid Mechanics,2014,43(3):292–301.(in Chinese))
[20]史文谱,魏娟.圆形区域多圆孔对SH波散射Green函数解[J].振动与冲击,2014,33(20):31–34.(SHI Wenpu,WEI Juan.Green function solution of the SH-wave scattering of several circular cavities in circular domain[J].Journal of vibration and shock,2014,33(20):31–34.(in Chinese))
[21]梁建文,丁美,杜金金.柱面SH波在地下圆形衬砌洞室周围散射解析解[J].地震工程与工程振动,2013,33(1):1–7.(LIANG Jianwen,DING Mei,DU Jinjin.Diffraction of cylindrical SH waves around circular lined cavity:analytical solution[J].Journal of Earthquake Engineering and Engineering Vibration,2013,33(1):1–7.(in Chinese))
[22]韩峰,王光政,陈翰.SH波对多个凸起与凹陷相连地形的散射问题研究[J].应用数学和力学,2013,34(4):355–363.(HAN Feng,WANG Guangzheng,CHEN Han.Research on Scattering of SH Waves on Multiple Hills and Canyons[J].Applied Mathematics and Mechanics,2013,34(4):355–363.(in Chinese))
[23]齐辉,张根昌,郭晶.SH波入射半空间双相介质界面附近圆形衬砌的动力分析[J].固体力学学报,2013,34(4):426–432.(QI Hui,ZHANG Genchang,GUO Jing.Dynamic analysis of the scattering of SH-wave by circular lining near bi-material interface in half-space[J].Chinese Journal of Solid Mechanics,2013,34(4):426–432.(in Chinese))
[24]杨在林,许华南,黑宝平.半空间椭圆夹杂与裂纹对SH波的散射[J].振动与冲击,2013,32(11):56–61.(YANG Zailin,XU Huanan,HEI Baoping.Interaction of elliptical and crack under incident SH-wave in a half-space[J].Journal of Vibration and Shock,2013,32(11):56–61.(in Chinese))
[2]陈志刚,刘殿魁.SH波冲击下浅埋任意圆形孔洞的动力分析[J].地震工程与工程振动,2004,24(4):32–36.(CHEN Zhigang,LIU Diankui.Dynamic response on a shallowly buried cavity of arbitrary shape impacted by vertical SH-wave[J].Earthquake Engineering Vibration,2004,24(4):32–36.(in Chinese))
[3]LEE V W,MANOOGIAN M E.Surface motion above an arbitrary shape underground cavity for incident SH wave[J].European Earthquake Engineering,1995,8(1):3–11.
[4]LEE V W,AMORNWONGPAIBUN A.Scattering of anti-plane(SH)waves by a semi-elliptical hill:I-Shallow hill[J].Soil Dynamic and Earthquake Engineering,2012,52:116–125.
[5]AMORNWONGPAIBUN A,LEE V W.Scattering of anti-plane(SH)waves by a semi-elliptical hill:II-Deep hill[J].Soil Dynamic and Earthquake Engineering,2012,52:126–137.
[6]LUO H,LEE V W,LIANG J W.Anti-plane(SH)waves diffraction by an underground semi-circular cavity:analytical solution[J].Earthquake Engineering and Engineering Vibration,2010,9(3):385–396.
[7]LEE V W,LUO H,LIANG J W.Diffraction of anti-plane SH waves by a semi-circular cylindrical hill with an inside concentric semi-circular tunnel[J].Earthquake Engineering and Engineering Vibration,2004,3(2):249–262.
[8]QI H,YANG J.Dynamic analysis for circular inclusion of arbitrary positions near interfacial crack impacted by SH-wave in half-space[J].European Journal of Mechanics/A Solids,2012,36:18–24.
[9]HASSAN A,SONG T S.Dynamic anti-plane analysis for two symmetrically interfacial cracks near circular cavity in piezoelectric bi-materials[J].Applied Mathematics and Mechanics,2014,35(10):1 261–1 270.
[10]杨在林,许华南,黑宝平.双相介质中椭圆孔洞与裂纹对SH波的散射[J].工程力学,2014,31(10):33–39.(YANG Zailin,XU Huanan,HEI Baoping.Interaction of an elliptical cavity and a crack near bimaterials interface under incident SH-waves[J].Engineering Mechanics,2014,31(10):33–39.(in Chinese))
[11]陈志刚.SH波在界面孔上散射的远场解[J].振动与冲击,2014,33(12):138–144.(CHEN Zhigang.Far field solution to SH-wave scattered by an interface cavity[J].Journal of Vibration and Shock,2014,33(12):138–144.(in Chinese))
[12]史文谱,陈瑞平,张春萍.直角平面内弹性圆夹杂对入射平面SH波的散射[J].应用力学学报,2007,24(1):154–160.(SHI Wenpu,CHEN Ruiping,ZHANG Chunping.Scattering of circular inclusion in right-angle plane to incident plane SH-wave[J].Chinese Journal of Applied Mechanics,2007,24(1):154–160.(in Chinese))
[13]齐辉,杨杰,李宏亮,等.含任意直线型裂纹的直角域中圆柱夹杂对SH波的散射[J].振动与冲击,2011,30(5):208–212.(QI Hui,YANG Jie,LI Hongliang,et al.Scattering of SH-wave by a cylindrical inclusion in right-angle plane with arbitrary beeline crack[J].Journal of Vibration and Shock,2011,30(5):208–212.(in Chinese))
[14]QI H,SHI Y,LIU D K.Interaction of a circular cavity and a beeline crack in right-angle plane impacted by SH-wave[J].Journal of Harbin Institute of Technology,2009,16(4):548–553.
[15]杨在林,闫培雷,刘殿魁.SH波对浅埋弹性圆柱及裂纹的散射与地震动[J].力学学报,2009,41(2):229–235.(YANG Zailin,YAN Peilei,LIU Diankui.Scattering of SH-waves and ground motion by an elastic cylindrical inclusion and a crack in half space[J].Chinese Journal of Theoretical and Applied Mechanics,2009,41(2):229–235.(in Chinese))
[16]宋天舒,李冬.压电体中孔边Ⅲ型界面裂纹的动应力强度因子[J].力学学报,2010,42(6):1 219–1 224.(SONG Tianshu,LI Dong.Dynamic stress intensity factor for interfacial cracks of mode III on a circular cavity in piezoelectric bimaterials[J].Chinese Journal of Theoretical and Applied Mechanics,2010,42(6):1 219–1 224.(in Chinese))
[17]杨在林,许华南,黑宝平.SH波上方垂直入射时界面附近椭圆夹杂与裂纹的动态响应[J].岩土力学,2013,34(8):2 378–2 384.(YANG Zailin,XU Huanan,HEI Baoping.Dynamic response of elliptical inclusion and crack neat interface under vertically incident SH-wave from above[J].Rock and Soil Mechanics,2013,34(8):2 378–2 384.(in Chinese))
[18]WANG J H,LU J F,ZHOU X L.Complex variable function method for scattering of plane waves by an arbitrary hole in a porous medium[J].European Journal of Mechanics/A Solids,2009,28:582–590
[19]刘中宪,刘蕾,梁建文.楔形空间中任意形状孔洞对平面SH波的散射:IBEM求解[J].固体力学学报,2014,43(3):292–301.(LIU Zhongxian,LIU Lei,LIANG Jianwen.Solving the scattering of plane SH-waves by an arbitrary shaped cavity embedded in a wedge-shaped space by IBEM[J].Chinese Journal of Solid Mechanics,2014,43(3):292–301.(in Chinese))
[20]史文谱,魏娟.圆形区域多圆孔对SH波散射Green函数解[J].振动与冲击,2014,33(20):31–34.(SHI Wenpu,WEI Juan.Green function solution of the SH-wave scattering of several circular cavities in circular domain[J].Journal of vibration and shock,2014,33(20):31–34.(in Chinese))
[21]梁建文,丁美,杜金金.柱面SH波在地下圆形衬砌洞室周围散射解析解[J].地震工程与工程振动,2013,33(1):1–7.(LIANG Jianwen,DING Mei,DU Jinjin.Diffraction of cylindrical SH waves around circular lined cavity:analytical solution[J].Journal of Earthquake Engineering and Engineering Vibration,2013,33(1):1–7.(in Chinese))
[22]韩峰,王光政,陈翰.SH波对多个凸起与凹陷相连地形的散射问题研究[J].应用数学和力学,2013,34(4):355–363.(HAN Feng,WANG Guangzheng,CHEN Han.Research on Scattering of SH Waves on Multiple Hills and Canyons[J].Applied Mathematics and Mechanics,2013,34(4):355–363.(in Chinese))
[23]齐辉,张根昌,郭晶.SH波入射半空间双相介质界面附近圆形衬砌的动力分析[J].固体力学学报,2013,34(4):426–432.(QI Hui,ZHANG Genchang,GUO Jing.Dynamic analysis of the scattering of SH-wave by circular lining near bi-material interface in half-space[J].Chinese Journal of Solid Mechanics,2013,34(4):426–432.(in Chinese))
[24]杨在林,许华南,黑宝平.半空间椭圆夹杂与裂纹对SH波的散射[J].振动与冲击,2013,32(11):56–61.(YANG Zailin,XU Huanan,HEI Baoping.Interaction of elliptical and crack under incident SH-wave in a half-space[J].Journal of Vibration and Shock,2013,32(11):56–61.(in Chinese))